MA233: Elementary Number Theory

Unit 2: The Greatest Common DivisorEven when an integer is not prime, it can still behave in a prime-like
fashion with many other integers. We call two integers that act this
wayrelatively prime. This concept is as important to elementary
number theory as the notion of prime numbers, and it depends entirely on
thegreatest common divisorof the two integers in question,
abbreviated as theirgcd. A surprisingly efficient method to
compute the gcd is due to the same Euclid who showed us earlier that
there are infinitely many primes, and we will study it here in some
detail.

We next turn to the topic of *linear Diophantine equations. These
are basically the same linear equations that you studied in precalculus,
but with a twist: we only want integer solutions. This greatly restricts
what we can do, but the gcd provides a criterion for when a linear
Diophantine equation can be solved, and the Euclidean algorithm – which
we need to compute the gcd anyway – provides a splendid technique for
solving them. This leads us to **Bezout’s identity, which “turns the
tables” on these relationships and provides a powerful tool for
subsequent units.*

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- define the greatest common divisor (gcd) and identify important
properties of the gcd;
- compute the gcd of two integers using the Euclidean Algorithm and
predict how many steps it will take;
- explain why there are infinitely many prime numbers in arithmetic
sequences with a certain property defined by the gcd;
- determine which linear Diophantine equations have solutions and
solve them; and
- use the Extended Euclidean Algorithm to write the gcd in terms of
Bezout’s identity.

After attempting the exercises assigned above, discuss your
solutions in the course discussion forum. Feel free to respond to
other students’ postings as well. If you haven’t already done so,
you will need to create a free account at the link above to
participate in the discussions.

Instructions: Reread “Section 2.4.1: The Fundamental Theorem of
Arithmetic” on pages 41–44. Then read “Section 2.4.2: More on the
Infinitude of Primes” on pages 45-46.
Reading these sections, taking notes, and studying the examples
should take approximately 1 hour.

After attempting the exercise assigned above, discuss your solution
in the course discussion forum. Feel free to respond to other
students’ postings as well. If you haven’t already done so, you will
need to create a free account at the link above to participate in
the discussions.

Instructions: Read “Section 1.6: The Euclidean Algorithm” on pages
25-28. Be sure to follow the examples carefully!
Reading this section, taking notes, and studying the examples
should take approximately 30 minutes.

After attempting the exercises assigned above, discuss your
solutions in the course discussion forum. Feel free to respond to
other students’ postings as well. If you haven’t already done so,
you will need to create a free account at the link above to
participate in the discussions.

After attempting the exercises assigned above, discuss your
solutions in the course discussion forum. Feel free to respond to
other students’ postings as well. If you haven’t already done so,
you will need to create a free account at the link above to
participate in the discussions.

After attempting the exercises assigned above, discuss your
solutions in the course discussion forum. Feel free to respond to
other students’ postings as well. If you haven’t already done so,
you will need to create a free account at the link above to
participate in the discussions.

Instructions: Try to do Exercise 7 on page 34.
After attempting the exercise assigned above, discuss your solution
in the course discussion forum. Feel free to respond to other
students’ postings as well. If you haven’t already done so, you will
need to create a free account at the link above to participate in
the discussions.
Completing this assessment should take approximately 1 hour.

After attempting the exercises assigned above, discuss your
solutions in the course discussion forum. Feel free to respond to
other students’ postings as well. If you haven’t already done so,
you will need to create a free account at the link above to
participate in the discussions.

Instructions: Download the linked set of labs. Upload the second
one (2.3.2-SageWS2.sws) to the Sage website where you created an
account (subunit 1.4.2). Work through the lab carefully.
Completing this assignment should take approximately 30 minutes.